Aula Seminari, Dipartimento di Matematica
In 1961 Rees proved a criterion for integral dependence between two ideals of finite colengths by the equality of their Hilbert-Samuel multiplicity. This criterion plays an important role in Teissier’s work on the equisingularity of families of hypersurfaces with isolated singularities. For hypersurfaces with non-isolated singularities, one needs a similar numerical criterion for integral dependence of arbitrary ideals. For a long time, it was not clear how to extend Rees’ multiplicity theorem to arbitrary ideals because they do not have the Hilbert-Samuel multiplicity. A possibility is to replace the Hilbert-Samuel multiplicity by the multiplicity sequence which were introduced by Achilles and Manaresi in 1997. Independent works of Gaffney and Gassler in the analytic case have led to the conjecture that two arbitrary ideals in a local ring as in Rees’ work have the same integral closure if and only if they have the same multiplicity sequence. This conjecture was recently solved by Claudia Polini, Ngo Viet Trung, Bernd Ulrich, and Javid Validashti. This talk will discuss the development leading to the solution and arising problems.